On the -Theory of Finite Fields

Home , Adams operation, Plus construction

ON THE 퐾-THEORY OF FINITE FIELDS

PETER J. HAINE

Abstract. In this talk we introduce higher algebraic 퐾-groups via Quillen’s plus construction. We then give a brief tour of algebraic 퐾-theory and its relation to stable homotopy theory and number theory — in particular noting that a 퐾-theory computation of the integers is equivalent to the Kummer–Vandiver conjecture. We next state Quillen’s computations of the 퐾-theory of 퐅푞 and 퐅푞. The remainder of the talk focuses on how Quillen is able to execute these 퐾-theory computations by using the Adams operations to relate 퐾-theory to modular representation theory. We finish by discussing the Brauer lift to — the main tool that makes Quillen’s computation go — and how Quillen uses the Brauer lift to show that + 푞 퐵GL(퐅푞) is homotopy equivalent to a space 퐹휓 , defined via the Adams operations, whose homotopy groups are easily computable using Bott periodicity.

Contents 1. Algebraic 퐾-theory via perfect groups and the Plus Construction 1 2. A Whirlwind Tour of 퐾-theory 4 3. The 퐾-theory of finite fields 4 4. The Adams Operations via 휆-rings 5 5. The Brauer Lift 8 6. The space 퐹휓푞 and computing the 퐾-theory of finite fields 9 References 11

1. Algebraic 퐾-theory via perfect groups and the Plus Construction In this section we define higher algebraic 퐾-theory via Quillen’s plus construction. To motivate this, we recall how the group 퐾1 was first defined via elementary matrices. 1.1. Convention. For us 푅 always denotes a commutative (unital) ring.

1.2. Recollection. Let 푛 be a positive integer, and consider the group GL푛(푅) of 푛 × 푛 matrices. Given 1 ≤ 푖, 푗 ≤ 푛 and 푟 ∈ 푅, let 훿푖,푗 by the 푛 × 푛 matrix whose only nonzero entry is 1 in the (푖, 푗) position, and let 휀푖,푗(푟) ≔ id푛 +푟훿푖,푗. The matrices 휀푖,푗(푟) are called elementary matrices. Let 퐸푛(푅) denote the subgroup of GL푛(푅) generated by the elementary matrices. The infinite general linear group is the colimit GL(푅) ≔ colim ( GL1(푅) GL2(푅) GL2(푅) ⋯ ) .

Date: Fall 2016. Acknowledgments: We thank Robin Elliott for watching a preliminary version of this talk and for providing extremely helpful suggestions on its contents and organization. Note: Items marked with a “ * ” are extra remarks that we likely did not have time to mention in the talk. 1 2 PETER J. HAINE

Similarly, define 퐸(푅) ≔ colim ( 퐸1(푅) 퐸2(푅) 퐸2(푅) ⋯ ) .

1.3. Remark. If 푅 is a field, then 퐸푛(푅) = SL푛(푅). 1.1. Preliminary Definition. Let 푅 be a commutative ring. The first algebraic 퐾-group of 푅 is the quotient

퐾1(푅) ≔ GL(푅)/퐸(푅) . As algebraic topologists, we prefer definitions to have a homotopy-theoretic flavor. In particular, we might try to define 퐾1(푅) in terms of homotopy groups. We already know that 휋1(퐵GL(푅)) ≅ GL(푅). Moreover, 퐵GL(푅) has an explicit CW-structure. One idea to produce 퐾1 homotopy–theoretically is to add cells to 퐵GL(푅) to create a space whose fundamental group is 퐾1(푅) — is is precisely what the plus construction will do for us. Before we do this, let us take a closer look at 퐸(푅) to see exactly what sort of group-theoretic properties 퐸(푅) has. 1.4. Notation. Let 퐺 be a group. Write [퐺, 퐺] for the commutator subgroup of 퐺 and write 퐺ab ≅ 퐺/[퐺, 퐺] for the abelianization of 퐺.

1.5. Example. For 푛 ≥ 3, the subgroup 퐸푛(푅) is a normal subgroup of GL푛(푅) with the property that [퐸푛(푅), 퐸푛(푅)] = 퐸푛(푅). Moreover, [퐸(푅), 퐸(푅)] = 퐸(푅).

1.6. Definition. A group 퐺 is perfect if 퐺ab = 0.

1.7. Example. The alternating group 퐴5 ⊲ 훴5 is the smallest nontrivial perfect group. 1.8. Example. The infinite symmetric group is the colimit 훴∞ ≔ colim ( 훴1 훴2 훴2 ⋯ ) . The infinite alternating group is the colimit 퐴∞ ≔ colim ( 퐴1 퐴2 퐴2 ⋯ ) .

The index of 퐴∞ in 훴∞ is 2, and 퐴∞ is a perfect normal subgroup of 훴∞. Acyclic spaces generate a class of examples of perfect groups. ̃ 1.9. Definition. A space 푋 is acyclic if 퐻∗(푋; 퐙) = 0.

1.10. Lemma. Let 푋 be an acyclic space. Then 푋 is connected and 휋1(푋) is perfect.

Proof. Since 푋 is acyclic, 퐻0(푋) ≅ 퐙, hence 푋 is connected. Similarly, since 푋 is acyclic 퐻1(푋) = 0, but the Hurewicz homomorphism exhibits 퐻1(푋) as an abelianization of 휋1(푋), □ hence 휋1(푋)ab = 0. 1.11. Definition. Let 푋 and 푌 be pointed connected CW-complexes. A pointed map 푓∶ 푋 → 푌 is acyclic if hofib(푓) is acyclic. 1.12. Lemma. Let 푋 and 푌 be connected CW-complexes. A map 푓∶ 푋 → 푌 is acyclic if and only if for every local coefficient system 퐿∶ 훱1(푌) → Ab, the map 푓 induces an isomorphism

푓⋆ ∶ 퐻∗(푋; 퐿 ∘ 훱1(푓)) ⥲ 퐻∗(푌; 퐿) . ON THE 퐾-THEORY OF FINITE FIELDS 3

Proof Sketch. The proof is not very hard — the main tool of is the Serre spectral sequence, and the comparison theorem for cellular maps of fibrations. A proof can be found in [13, Ch. 4 Lem. 1.6]. □

1.13. Definition. Let 푋 be a pointed connected CW-complex and 푃 ⊲ 휋1(푋) a perfect normal subgroup. An acyclic map 휄∶ 푋 → 푋+ is called a plus construction for 푋 (relative + to 푃) if 푃 is the kernel of 휄⋆ ∶ 휋1(푋) → 휋1(푋 ).

1.14. Theorem (Quillen). Let 푋 be a pointed CW-complex and 푃 ⊲ 휋1(푋) a perfect normal subgroup. Then there exists a plus construction 휄∶ 푋 → 푋+ for 푋 relative to 푃. Moreover, the plus construction enjoys the following universal property: for every map 푓∶ 푋 → 푌 so that + + 푓⋆(푃) = 0, there is a map 푓 ∶ 푋 → 푌, unique up to pointed homotopy, so that the triangle

푋 휄 푋+ ≃ 푓 푓+ 푌 commutes up to pointed homotopy.

Idea of the construction of 푋+. The idea of the construction of 푋+ is to attach 2-cells to 푋 to kill 푃 from 휋1(푋), then attach 3-cells to make sure that we retain the correct homology. Proofs can be found in [6, Prop. 4.40; 9, Thé. 1.1.1]. From this explicit construction, it is possible to use obstruction theory to show that 푋+ has the desired universal property, which in turn characterizes 푋+ uniquely up to homotopy equivalence. □

The plus construction is functorial in the following sense.

perf 1.15. Corollary. Let CW∗ denote the category with objects pairs (푋, 푃), where 푋 is a pointed connected CW-complex and 푃 ⊲ 휋1(푋) is a perfect normal subgroup, and a morphism 푓∶ (푋, 푃) → (푋′, 푃′)

′ ′ is a map 푓∶ 푋 → 푋 so that 푓⋆(푃) ⊂ 푃 . The plus construction defines a functor

+ perf (−) ∶ CW∗ → ℎCW∗ . 1.16. Notation. For a commutative ring 푅, let 퐵GL(푅)+ denote the plus construction of 퐵GL(푅) with respect to the perfect normal subgroup 퐸(푅) ⊲ GL(푅) ≅ 휋1(퐵GL(푅)). 1.17. Definition. Let 푅 be a commutative ring and 푛 a positive integer. The 푛th algebraic 퐾-group of 푅 is the homotopy group

+ 퐾푛(푅) ≔ 휋푛(퐵GL(푅) ) . 1.18. Example. By the definition of the plus construction, for any ring 푅 we have

퐾1(푅) ≅ GL(푅)/퐸(푅) , so this definition agrees with the preliminary definition of 퐾1.

1.19. Warning. Notice that we have expressly avoided mentioning anything about 퐾0! That is because 퐾0 needs to be defined differently — Jesse will give a more careful analysis of 퐾0, 퐾1, and 퐾2 in his talk on 3 October. 4 PETER J. HAINE

2. A Whirlwind Tour of 퐾-theory In this section we give one manifestation of the connections between algebraic 퐾-theory and stable homotopy theory, via the Barratt–Priddy–Quillen theorem, as well as indicate how difficult 퐾-theory computations are by stating the the Kummer–Vandiver conjecture from number theory is equivalent to a 퐾-theory computation of the integers. + 2.1. Theorem (Barratt–Priddy–Quillen [3]). Let 퐵훴∞ denote the plus construction of 퐵훴∞ with respect to 퐴∞. There is an equivalence + ∞ ∞ 0 푛 푛 0 푛 푛 퐵훴∞ ≃ 훺 훴 푆 = colim푛 훺 훴 푆 = colim푛 훺 푆 . + 푆 In particular, 퐵훴∞ ≅ 휋푛.

*2.1. Philosophical Remark. Now, 훴∞ is not the infinite general linear group of any ring, but if there were a field 퐅1 with one element, heuristically GL푛(퐅1) should be 훴푛, so GL(퐅1) 푆 should be 훴∞. Then under this heuristic, 퐾푛(퐅1) ≅ 휋푛. This is only philosophy, but later we will see that this can be made precise with the algebraic 퐾-theory of various types of 푆 categories. In this setting 퐾푛(Fin∗) ≅ 휋푛. 2.2. Observation (Relating algebraic 퐾-theory to stable homotopy theory). For each positive integer 푛, there is an inclusion 훴푛 ↪ GL푛(퐙) given by the permutation representation, hence this induces an inclusion 훴∞ ↪ GL(퐙). The elementary integer matrices generate SL푛(퐙), and the matrix of an even permutation is 1, so we have compatible inclusions

퐴∞ SL(퐙)

훴∞ GL(퐙) , + + so by the functoriality of the plus construction we get an induced map 퐵훴∞ → 퐵GL(퐙) . Taking homotopy groups and using the Barratt–Priddy–Quillen theorem gives a map 푆 휋푛 → 퐾푛(퐙) .

2.3. Observation. Since 퐙 is the initial ring, every ring receives a unique map 휙푅 ∶ 퐙 → 푅. Since taking the general linear group is a functor and the image of an elementary matrix is an elementary matrix, by the universal property of the plus construction, 휙푅 induces map + + 퐵GL(퐙) → 퐵GL(푅) , and applying homotopy groups yields a map 퐾푛(퐙) → 퐾푛(푅). Hence by Observation 2.2, stable homotopy maps to the 퐾-theory of every ring.

2.4. Proposition (Kurihara [8]). We have that 퐾4푛(퐙) = 0 for all positive integers 푛 if and only if the Kummer–Vandiver conjecture holds. 2.5. Conjecture (Kummer–Vandiver). For all primes 푝, the prime 푝 does not divide the class th number of the maximal real subfield of the 푝 cyclotomic field 퐐(휁푝). 2.6. Remark. The 퐾-theory of 퐙 is hair-raisingly difficult to compute — a summary of many of the known computations can be found in [13, Ch. 6 §10].

3. The 퐾-theory of finite fields Our goal is to understand the main ideas of the proof of the following amazing computation of Quillen. 3.1. Theorem (Quillen). Let 푞 be a power of a prime 푝. 푛 (3.1.a) For all 푛 ≥ 1 we have 퐾2푛(퐅푞) = 0 and 퐾2푛−1(퐅푞) ≅ 퐙/(푞 − 1). ON THE 퐾-THEORY OF FINITE FIELDS 5

(3.1.b) If 휄∶ 퐅푞 ↪ 퐅푞푚 is a field extension, then 휄⋆ ∶ 퐾∗(퐅푞) → 퐾∗(퐅푞푚 ) is injective. 푝 (3.1.c) Let Fr ∶ 퐅푞 ⥲ 퐅푞 denote the absolute Frobenius automorphism. Then for all 푛 ≥ 1, the induced homomorphism 푝 Fr⋆ ∶ 퐾2푛−1(퐅푞) → 퐾2푛−1(퐅푞) is given by multiplication by 푝푛. 3.2. Corollary (Quillen). Let 푝 be a prime.

(3.1.1.a) For all 푛 ≥ 1 we have 퐾2푛(퐅푝) = 0 and

퐾2푛−1(퐅푝) ≅ ⨁ 퐐ℓ/퐙ℓ . ℓ≠푝 prime

(3.1.1.b) The automorphism of 퐾2푛−1(퐅푝) induced by the Frobenius automorphism is given by multiplication by 푝푛. (3.1.1.c) If 퐅푞 ⊂ 퐅푝 is a subfield, then the canonical map

Gal(퐅푝/퐅푞) 퐾∗(퐅푞) → 퐾∗(퐅푝) is an isomorphism.

4. The Adams Operations via 휆-rings In this section we recall the properties of the Adams operations on 퐾-theory in the setting of 휆-rings as the representation ring of a finite group also has the structure of a 휆-ring, and in both cases Adams operations can be constructed using only the 휆-ring structure, as well as an additional augmentation given by the dimension. We motivate 휆-rings by considering the extra structure on VB퐂(푋) given by taking exterior powers. ′ 4.1. Observation (Extra structure on VB퐂(푋)). Notice that if 퐸 and 퐸 are complex vector bundles over a finite CW-complex 푋, we can perform the following constructions, which give extra structure to VB퐂(푋). (4.1.a) 훬0(퐸) = 푋 × 퐂. (4.1.b) 훬1(퐸) = 퐸. 푛 ′ 푖 푗 ′ (4.1.c) 훬 (퐸 ⊕ 퐸 ) ≅ ⨁푖+푗=푛 훬 (퐸) ⊗ 훬 (퐸 ). However, notice that these exterior power operations are not semiring homomorphisms.

4.2. Observation. Let 퐺 be a finite group and 푘 a field. Notice that the set Rep푘(퐺) of isomorphism classes of finite-dimensional 푘-representations of 퐺 has the structure of a commutative semiring, with the additive structure given by direct sum and the multiplicative structure given by tensor product — this structure is analogus to the extra structure on VB퐂(푋). 4.3. Definition. A 휆-structure on a commutative semiring 푆 is a collection of set-maps 푛 {휆 ∶ 푆 → 푆}푛≥0 called 휆-operations so that (4.3.a) 휆0 is the constant map at the identity 1 of 푆. 1 (4.3.b) 휆 = id푆. × 푛 푛 (4.3.c) The map 휆푡 ∶ 푆 → 푆⟦푡⟧ given by 휆푡(푠) ≔ ∑푛≥0 휆 (푠)푡 is a monoid map. We call a semiring equipped with a 휆-structure a 휆-semiring.A 휆-ring is just a ring that is also a 휆-semiring. 6 PETER J. HAINE

4.4. Remark. Condition (4.3.c) is equivalent to saying that for all 푠, 푠′ ∈ 푆 we have 휆푛(푟 + 푟′) = ∑ 휆푖(푠)휆푗(푠′) , 푖+푗=푛 which is an abstraction of (4.1.c). 4.5. Remark (on terminology). The terminology in the literature on 휆-rings is quite incon- sistent. Some call what we have called a 휆-ring a pre-휆-ring, and insist that 휆-rings satisfy three additional properties. Two of these properties encode the expressions of exterior powers 훬푛(훬푛(푉)) and 훬푛(푉 ⊗ 푉′) of vector spaces 푉 and 푉′ as integral polynomials in the exterior powers 훬1(푉), …, 훬푚푛(푉) and 훬1(푉), …, 훬푛(푉), 훬1(푉′), …, 훬푛(푉′), respectively. The other property is that 휆푘(1) = 0 if 푘 > 1, which encodes the fact that 훬푘(퐿) = 0 for 푘 > 1 if 퐿 is a 1-dimensional vector space. Others take our convention and call a 휆-ring with this extra structure a special 휆-ring. We have chosen this approach to simplify terminology, though we actually agree that a 휆-ring really should satisfy these additional properties. Moreover, all of the 휆-rings that we need to consider do satisfy these properties. 4.6. Remark. The 휆-operations in some sense are very unnatural as they are not ring homomorphisms. One might ask where these operations come from — the answer is plethories and the plethistic algebra of Borger and Wieland [4].

4.7. Example. The semirings VB퐂(푋) and Rep푘(퐺) are both 휆-semirings with the 휆-operations given by taking exterior powers. 4.8. Example. The nonnegative integers 퐍 are a 휆-semiring with 휆-operations given by 푛 휆푘(푛) = ( ) . 푘 4.9. Example. A 퐐-algebra 퐴 is a 휆-ring with 휆-operations given by 1 휆푘(푎) = 푎(푎 − 1) ⋯ (푎 − 푘 − 1) . 푘! 4.10. Notation. Given a commutative monoid 푀, write 푀gp for the group completion of 푀. 4.11. Lemma. If 푆 is a 휆-semiring. Then 푆gp is naturally a 휆-ring 휆-operations extending the 휆-operations on 푆. × gp × Proof. Since the map 휆푡 ∶ 푆 → 푆⟦푡⟧ is a monoid map and 푆 ⟦푡⟧ is a group completion for 푆⟦푡⟧×, by the universal property of the group completion, there exists a unique monoid gp map 휆푡 making the square 휆 푆 푡 푆⟦푡⟧×

gp gp × 푆 gp 푆 ⟦푡⟧ 휆푡 gp gp commute. The coefficents of 휆푡 define a 휆-ring structure on 푆 extending the 휆-ring structure on 푆. □ 4.12. Example. Applying Lemma 4.11 to Example 4.8, we see that the integers 퐙 are a 휆-ring with 휆-operations given by 푛 1 휆푘(푛) = ( ) = 푛(푛 − 1) ⋯ (푛 − 푘 − 1) . 푘 푘! ON THE 퐾-THEORY OF FINITE FIELDS 7

4.13. Remark (Witt vectors). The abelian group 푅⟦푡⟧× = 1 + 푡푅⟦푡⟧ naturally has the structure of a commutative ring, which we will not describe here, and is often called the ring of big Witt vectors, and denoted by 퐖(푅). Moreover, 퐖(푅) is naturally a 휆-ring, and the condition that a 휆-ring 푅 be a special 휆 ring can be concisely phrased by requiring that the group homomorphism 푅 → 퐖(푅) be a 휆-ring homomorphism. 4.14. Example. If 푋 is a finite CW-complex, then 퐾0(푋) is a 휆-ring with the 휆-structure given by taking exterior powers. 4.15. Example. Let 퐺 be a finite group and 푘 a field. The representation ring of 퐺 is the gp group completion 푅푘(퐺) ≔ Rep푘(퐺) . Hence 푅푘(퐺) is a 휆-ring. An element of 푅푘(퐺) is called a virtual representation of 퐺. 4.16. Remark. The category of 휆-rings has objects 휆-rings, and a morphism 휙∶ 푅 → 푅′ is 푛 푛 a ring homomorphism so that 휆푅′ ∘ 휙 = 휙 ∘ 휆푅 for all 푛 ≥ 0. 4.17. Definition. Let 푅 be a 휆-ring. An augmentation of 푅 is a 휆-subring 퐴 ⊂ 푅 equipped with a retraction of 휆-rings 휀∶ 푅 → 퐴 so that 휀|퐴 = id퐴. 0 4.18. Example. Both 퐾 (푋) and 푅퐂(퐺) have augmentations to 퐙 given by taking (virtual) dimension, induced by the morphisms 휆-semirings VB퐂(푋) → 퐍 and 푅퐂(퐺) → 퐍 given 0 by taking actual dimension. Explicitly, the 휆-subrings of 퐾 (푋) and 푅퐂(퐺) isomorphic to 퐙 providing the augmentation are simply the subrings given by integer multiples of the identity in each case (i.e., the trivial virtual bundles and trivial virtual representations, respectively). The fact that these are actually 휆-ring homomorphisms follows from the dimension formula for exterior powers: dim(푉) dim(훬푘(푉)) = ( ) . 푘 4.19. Definition (Adams operations). Let (푅, 휀) be an augmented 휆-ring. There exist natural 푘 set-maps 휓 ∶ 푅 → 푅 for all 푘 ≥ 0, called Adams operations, defined as follows. Let 휓푡 denote the formal power series defined by 푑 휓 (푟) ≔ 휀(푟) − 푡 log(휆 (푟)) . 푡 푑푡 −푡 푘 푘 Then 휓 (푟) is the coefficent of 푡 in 휓푡(푟). 4.20. Theorem. For any finite CW-complex 푋 and any finite group 퐺, the Adams operations 0 on 퐾 (푋) and 푅퐂(퐺) satisfy the following properties. (4.20.a) For all 푘 ≥ 0, the set-map 휓푘 is a ring homomorphism. (4.20.b) For all 푘, ℓ ≥ 0 we have 휓푘휓ℓ = 휓푘ℓ. (4.20.c) If 퐿 is a one dimensional (actual) vector bundle or representation, then 휓푘(퐿) = 퐿⊗푘. Idea of the proof. In the 퐾-theory case, the idea is to use the splitting principle for 퐾-theory. As it turns out, there is also a splitting principle for 푅퐂(퐺) [2, Thm. 1.5(ii)] — the fundamental observation is to identify 푅퐂(퐺) with the ring of complex virtual characters. There is a notion of a splitting principle for 휆-rings, and if this is satisfied, then (4.20.a) and (4.20.b) hold [13, Ch. 2 Prop. 4.4] The last condition is obviously something more particular to these settings where we have a notion of dimension. □

4.21. Remark. The Adams operations on 푅퐂(퐺) induce operations on the characters given by 휓푘(휒(푔)) = 휒(푔푘), where 휒 is a character and 푔 ∈ 퐺. 8 PETER J. HAINE

4.22. Observation. By the naturality of the Adams operation, 휓푘 restricts to an operation 퐾̃0 → 퐾̃0 which we also denote by 휓푘 4.23. Recollection. Let 푛 be a nonnegative integer. Then 퐾̃0(푆2푛) ≅ 퐙. In our computation of the 퐾-theory of finite fields we need the following fact about the Adams operations. 4.24. Lemma. Let 푛 be a nonnegative integer. The Adams operation 휓푘 ∶ 퐾̃0(푆2푛) → 퐾̃0(푆2푛) is given by multiplication by 푘푛. Proof sketch. The proof is elementary — for 푛 = 1, it follows immediately by looking at the canonical generator of 퐾̃0(푆2), and the general case follows by induction using Bott periodicity. A complete argument can be found in [7, Prop. 2.21]. □

5. The Brauer Lift Now we explain and exploit an incredible relation between 퐾-theory and representation theory of finite groups via the Adams operations. Throughout 퐺 denotes a finite group, but we are particularly interested in the case when 퐺 = GL푛(퐅푞). 5.1. Definition. Let 퐺 be a finite group. Define a ring homomorphism 0 휙퐺 ∶ 푅퐂(퐺) → 퐾 (퐵퐺) first on Rep퐂(퐺) by sending an isomorphism class [푉] of representations to the class of the balanced product 퐸퐺 ×퐺 푉, and then extending to 푅퐂(퐺) via the universal property of the group completion. 0 5.2. Observation. Since the 휆-operations are defined on 푅퐂(퐺) and 퐾 (퐵퐺) via taking exterior powers, it is not hard to believe that 휙퐺 is a morphism of 휆-rings, and indeed this is 0 true. This morphism of 휆-rings respects the augmentations of 푅퐂(퐺) and 퐾 (퐵퐺) defined in Example 4.18, so by the naturality of the Adams operations, for all 푞 ≥ 0, the square

휙퐺 0 푅퐂(퐺) 퐾 (퐵퐺)

휓푞 휓푞

0 푅퐂(퐺) 퐾 (퐵퐺) 휙퐺 commutes. Composing with the homomorphism 퐾0(퐵퐺) ≅ [퐵퐺, 퐙 × 퐵푈] → [퐵퐺, 퐵푈] ≅ 퐾̃0(퐵퐺) induced by the inclusion 퐵푈 ↪ 퐙 × 퐵푈 as {0} × 퐵푈, we get a homomorphism ̃0 푅퐂(퐺) → 퐾 (퐵퐺) , again natural with respect to the Adams operations.

× × 5.3. Convention. Fix, once and for all, an embedding 휌∶ 퐅푞 ↪ 퐂 . 5.4. Remark. Everything from here on will depend on 휌. This dependence will not be prob- lematic and is well understood through the lens of étale cohomology.

5.5. Notation. Let 퐺 be a finite group and 푉 a finite-dimensional 퐅푞-representation of 퐺. For an element 푔 ∈ 퐺, write 푆푉(푔) for the set of eigenvalues of 푔. Given an eigenvalue 휆 ∈ 푆푉(푔), write 휇(휆) for the multiplicity of 휆. ON THE 퐾-THEORY OF FINITE FIELDS 9

5.6. Definition. Let 퐺 be a finite group and 휒1, …, 휒푛 its distinct irreducible complex characters. A virtual character 휒∶ 퐺 → 퐂 is a class function so that 휙 = 푚1휒1 + ⋯ + 푚푛휒푛 for some 푚1, …, 푚푛 ∈ 퐙.

5.7. Definition. Let 퐺 be a finite group and 푉 a finite-dimensional 퐅푞-representation of 퐺. The Brauer character of 푉 is the function 휒푉 ∶ 퐺 → 퐂 defined by

휒푉(푔) ≔ ∑ 휇(휆)휌(휆) . 휆∈푆푉(푔)

5.8. Theorem (Green). Let 퐺 be a finite group and 푉 a finite-dimensional 퐅푞-representation of 퐺. The Brauer character 휒푉 is the character of a unique (up to isomorphism) virtual complex representation 푉br, called the Brauer lifting of 푉.

5.9. Observation. The Brauer character 휒푉 is clearly additive in 푉, that is 휒푉⊕푊 = 휒푉 + 휒푊, hence the Brauer lifting is additive as well: (푉 ⊕ 푊)br ≅ 푉br ⊕ 푊br. Thus the Brauer lifting defines a homomorphism (−)br ∶ 푅 (퐺) → 푅 (퐺) . 퐅푞 퐂 5.10. Observation. Extension of scalars defines a homomorphism

− ⊗ 퐅 ∶ 푅 (퐺) → 푅 (퐺) . 퐅푞 푞 퐅푞 퐅푞 For 푉 ∈ 푅 (퐺), write 퐅푞 푉 ≔ 푉 ⊗ 퐅 . 퐅푞 푞

Since 푉 is a representation over 퐅푞, the eigenvalues of 푔 acting on 푉 are stable under the Frobenius automorphism 푥 ↦ 푥푞. Then since 휓푞 acts on characters by 휓푞휒(푔) = 휒(푔푞), the Adams operation on 푅퐂(퐺) fixes the Brauer lift of 푉, hence we get a homomorphism

푞 (−)br ∘ (− ⊗ 퐅 )∶ 푅 (퐺) → 푅 (퐺)휓 , 퐅푞 푞 퐅푞 퐂 called the Brauer lift. Composing this with the homomorphism from Observation 5.2 and taking fixed points, we have produced a map

푞 푞 훽 ∶ 푅 (퐺) → 퐾̃0(퐵퐺)휓 ≅ [퐵퐺, 퐵푈]휓 . 퐺 퐅푞

6. The space 퐹휓푞 and computing the 퐾-theory of finite fields 6.1. Convention. From now on, let 푞 be a fixed nonnegative integer. (We really only care about the case that 푞 is a prime power.)

We have the Adams operations 휓푞 ∶ 퐾̃0 → 퐾̃0 on complex 퐾-theory. Since 퐾̃ is represented by 퐵푈, we would like these to be represented by endomorphisms of 퐵푈 by just applying the Yoneda lemma, but there is a slight problem. The space 퐵푈 is not a finite CW- complex, and 퐾-theory is only defined for finite CW-complexes. However, by analyzing an explicit cell structure on 퐵푈 and using the Milnor exact sequence, it is possible to show that 푛-ary operations on 퐾̃ are representable by maps 퐵푈푛 → 퐵푈.

6.2. Notation. We abusively write 휓푞 ∶ 퐵푈 → 퐵푈 for a map representing the Adams operation 휓푞 on 퐾̃0. 10 PETER J. HAINE

6.3. Definition. The space 퐹휓푞 is the pullback

퐹휓푞 ℎ Map(퐼, 퐵푈) ⌟

휙 (ev0,ev1)

퐵푈 푞 퐵푈 × 퐵푈 . (id퐵푈,휓 ) Hence a point the space 퐹휓푞 can be viewed as a pair (푥, 훾), where 푥 ∈ 퐵푈 and 훾 is a path from 푥 to 휓푞(푥). In this sense 퐹휓푞 is a homotopy-theoretic version of the space of fixed points of 휓푞. 6.4. Recollection. Recall that a simple space is a space with abelian fundamental group and the action of the fundamental group on the higher homotopy groups is trivial. 6.5. Theorem (Dror [5, Ex. 4.2]). If 푋 and 푌 are simple spaces and 푓∶ 푋 → 푌 induces an isomorphism on integral homology, then 푓 is a weak homotopy equivalence. 6.6. Remark. Peter May has a nice exposition on this that appeared in a collection of works celebrating Hilton’s 60th birthday [10], and is freely available on his website. 6.7. Goal. The goal is to construct a map 푞 휃∶ 퐵GL(퐅푞) → 퐹휓 that induces an isomorphism on integral homology. The universal property of the plus con- + + 푞 struction will then give a map 휃 ∶ 퐵GL(퐅푞) → 퐹휓 , which is again an isomorphism on + 푞 integral homology. After showing that 퐵GL(퐅푞) and 퐹휓 are simply, a simple application of Theorem 6.5 and Whitehead’s theorem shows that 휃+ is a homotopy equivalence. 6.8. Lemma. We have that 퐹휓푞 ≃ hofib(1 − 휓푞). 푞 푞 6.9. Lemma. The space 퐹휓 is simple and we have 휋2푛(퐹휓 ) = 0 and 푞 푛 휋2푛−1(퐹휓 ) = 퐙/(푞 − 1) . Proof sketch. By Lemma 6.8, we have a long exact sequence in homotopy

푞 ⋆ (1−휓 ) 휕 푞 ⋯ 휋푚(퐵푈) 휋푚(퐵푈) 휋푚−1(퐹휓 ) 휋푚−1(퐵푈) ⋯ . ̃0 2푛 By Bott periodicity, 휋2푛−1(퐵푈) = 0 and 휋2푛(퐵푈) = 퐾 (푆 ) ≅ 퐙. In the latter case, by Lemma 4.24, the map 1 − 휓푞 is given by multiplication by 1 − 푞푛. This immediately gives the computations of the homotopy groups of 퐹휓푞. The fact that 퐹휓푞 is simple follows immediately from a general result about fibrations, using the fact that 퐵푈 is simply connected. □

6.10. Lemma. Suppose that 푋 is a CW-complex so that [푋, 푈] = 0. Then 푞 휓푞 휙⋆ ∶ [푋, 퐹휓 ] → [푋, 퐵푈] is an isomorphism. Proof sketch. This lemma follows almost immediately from the universal property of the pullback 퐹휓푞. □

푞 6.11. Lemma. Suppose that 퐺 is a finite group. Then [퐵퐺, 퐵푈]휓 ≅ [퐵퐺, 퐹휓푞]. ON THE 퐾-THEORY OF FINITE FIELDS 11

The proof of Lemma 6.11 follows almost immediately from the Atiyah–Segal completion theorem [1, Thm. 2.1] and Lemma 6.10. However, given what we have covered, stating the Atiyah–Segal completion theorem is a bit beyond the scope of the lecture, so we will just take Lemma 6.11 for granted. 푞 6.12. Construction. We construct the map 휃 by constructing maps 휃푛 ∶ 퐵GL푛(퐅푞) → 퐹휓 , then passing to the colimit. To construct the maps 휃푛, we use the Brauer lift with 퐺 = GL푛(퐅푞). By Lemma 6.11, the Brauer lift gives a map 푞 훽 ∶ 푅 (GL (퐅 )) → [퐵GL (퐅 ), 퐵푈]휓 ≅ [퐵GL (퐅 ), 퐹휓푞] . GL푛(퐅푞) 퐅푞 푛 푞 푛 푞 푛 푞 푞 Let 휃푛 ∶ 퐵GL푛(퐅푞) → 퐹휓 be a map representing the homotopy class given by the image of the standard representation of GL푛(퐅푞) on 퐅푞. These maps 휃푛 are compatible the with formation of the colimit colim푛 퐵GL푛(퐅푞), hence determine a homotopy class of maps 푞 휃∶ 퐵GL(퐅푞) → 퐹휓 . Moreover, since [퐵GL(퐅푞), 퐵푈] ≅ lim푛[퐵GL푛(퐅푞), 퐵푈], this class is unique. The bulk of Quillen’s paper [12, §§2–6, §8, §9, §11] is dedicated to computing the homol- 푞 ogy and cohomology of 퐵GL(퐅푞) and 퐹휓 with various coefficients to show that the map 휃 is an integral homology equivalence.

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